This appendix shows how the (-1)k factors arise in Eq. (7-13) for an even-N multisection linear-phase complex FSF. Substituting the positive-frequency, 0 k (N/2)–1, |H(k)|ejf(k) gain factors, with f(k) phase values from Eq. (7-11), into Eq. (7-10) gives

where the subscript "pf" means positive frequency. Focusing only on the numerator inside the summation in Eq. (G-16), it is

Equation G-17

showing how the (–1)k factors occur within the first summation of Eq. (7-13). Next we substitute the negative-frequency |H(k)|ejf(k) gain factors, (N/2)+1 k N–1, with f(k) phase values from Eq. (7-11''), into Eq. (7-10) giving

Equation G-18

where the subscript "nf" means negative frequency. Again, looking only at the numerator inside the summation in Eq. (G-18), it is

Equation G-19

That ejpN factor in Eq. (G-19) is equal to 1 when N is even, so we write

Equation G-20

establishing both the negative sign before, and the (–1)k factor within, the second summation of Eq. (7-13). To account for the single-section for the k = N/2 term (this is the Nyquist, or s/2, frequency, where w = p) we plug the |H(N/2)|ej0 gain factor, and k = N/2, into Eq. (7-8) giving

Equation G-21


Prev don't be afraid of buying books Next

Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183 © 2008-2020.
If you may any questions please contact us: